A transformation changes a parent function’s graph in a predictable way. Adding k outside, f(x)+k, shifts the graph up or down. Replacing x with x+k shifts it left or right. Multiplying by k, k·f(x), stretches, shrinks or flips it. Absolute value, piecewise and step functions combine these ideas.
Key facts:
- Standards F-BF.B.3 and F-IF.C.7b: describe how f(x)+k, f(x+k) and kf(x) change a graph, and graph absolute value, piecewise-defined and step functions.
- Outside changes act on y and are intuitive; inside changes act on x and work backward: x – 4 moves right 4.
- The graph of a|x – h| + k has its vertex at (h, k) and opens up when a > 0 and down when a < 0.
- On a piecewise graph, a closed circle includes the endpoint and an open circle excludes it.
- A step function has horizontal pieces and jumps, like a price charged per box or per hour or part of an hour.
What is a transformation of a function?
A transformation changes the graph of a parent function in a predictable way. Algebra 1 uses three parent functions: linear \(f(x)=x\), quadratic \(f(x)=x^{2}\) and absolute value \(f(x)=|x|\). Each change to the equation causes one matching change in the graph.
| New function | Effect on the graph |
|---|---|
| \(f(x)+k\) | Vertical shift: up \(k\) units if \(k>0\), down \(|k|\) units if \(k < 0\) |
| \(f(x+k)\) | Horizontal shift: left \(k\) units if \(k>0\), right \(|k|\) units if \(k < 0\) |
| \(k\cdot f(x)\) | Vertical stretch if \(|k|>1\), shrink if \(0 < |k| < 1\), flip over the x-axis if \(k < 0\) |
How does f(x + k) move a graph?
A number added inside the parentheses changes the input before the function acts, so the shift is horizontal and works backward. For \(f(x)=|x|\), the function \(f(x-4)=|x-4|\) equals 0 when \(x=4\), so its vertex moves to \((4,0)\): 4 units right. Likewise \(|x+2|\) equals 0 at \(x=-2\), so it moves 2 units left.
What does k times f(x) do?
Multiplying the output by \(k\) scales every y-value. For \(f(x)=|x|\), the point \((1,1)\) becomes \((1,2)\) on \(2|x|\), \((2,1)\) on \(\frac{1}{2}|x|\) and \((1,-1)\) on \(-|x|\). Points on the x-axis never move, because \(k\cdot 0=0\).
How do you graph an absolute value function?
Write the function as \(g(x)=a|x-h|+k\). The vertex is \((h,k)\). The graph opens up if \(a>0\) and down if \(a < 0\), and its two branches have slopes \(a\) and \(-a\).
- Plot the vertex \((h,k)\).
- Move 1 unit right and \(a\) units up (down if \(a\) is negative) to plot a second point.
- Mirror that point across the vertical line \(x=h\).
- Draw two rays from the vertex through these points.
Worked example. Graph \(g(x)=-2|x+1|+3\). Here \(h=-1\), \(k=3\) and \(a=-2\). The vertex is \((-1,3)\). One unit right gives \((0,1)\), and its mirror is \((-2,1)\).
What are piecewise and step functions?
A piecewise function uses a different rule on each part of its domain. For example, \(f(x)=x+2\) when \(x < 1\), and \(f(x)=4-2x\) when \(x\ge 1\). At the boundary, a closed circle marks a point that belongs to the graph and an open circle marks a point that does not.
A step function is a piecewise function whose pieces are horizontal segments. It models prices that jump, such as $4 for each box of up to 6 cookies. The cost of 7 cookies is \(\$8\), not \(\$4.67\).
Transformations, Absolute Value and Piecewise Functions: frequently asked questions
Why does f(x – 3) move the graph to the right?
The output f(x – 3) reaches a given value when the input is 3 larger than before. The vertex of (x – 3)^2 occurs when x – 3 = 0, which is x = 3, so every point shifts 3 units right.
What is the difference between f(x) + k and f(x + k)?
f(x) + k adds k to every output and shifts the graph vertically. f(x + k) adds k to every input and shifts the graph horizontally in the opposite direction. For f(x) = x^2 and k = 3, the vertex moves to (0, 3) in the first case and to (-3, 0) in the second.
How do you find the vertex of an absolute value function?
Write the function as a|x – h| + k. The vertex is the point (h, k). For y = 3|x + 5| – 2, h = -5 and k = -2, so the vertex is (-5, -2).
What does an open circle mean on a piecewise graph?
An open circle means the graph approaches that point but does not include it. The value at that x-coordinate is given by another piece, usually shown with a closed circle.
How is a step function different from a linear function?
A linear function changes steadily and has a graph that is one connected line. A step function stays constant on an interval and then jumps, so its graph looks like stairs and has gaps between pieces.
- Transformations, Absolute Value and Piecewise Functions worksheet (24 problems)
- Transformations, Absolute Value and Piecewise Functions quiz (10 questions)
- Transformations, Absolute Value and Piecewise Functions test (45 minutes)
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See every chapter on the Algebra 1 math page.